It can be verified that \(\Gamma \)  as above is a disjoint union of trees. In this case, \(\mathcal {D}_i(\Gamma ,\sigma )\)  counts the total number of islands formed out of all possible induced subgraphs of \(\Gamma \)  with \(i\)  number of vertices. This count was first used to measure the multi-partite information of a topologically ordered system [1]}, [2]}, [3]} in theoretical condensed matter physics, where the physical attributes of a subgraph \(\Gamma ^{\prime }\)  were dependent on the number of islands of a subgraph.
